Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Friday, February 26, 2010

Is Math too hard?

Are humans good at Math?

In the movie
Oh God Book II
God (played by George Burns) says that
Math was a mistake, I made it too hard!.
While I am reluctant to contradict God, George Burns, or God as portrayed by George Burns,
scientists have found evidence that people are pretty good
at math. At least people have known about numbers for a far
longer time than previously thought.
See
this article.
What does this mean for us? The next time one of your students
says I can't do this! I'm just not that good at math!
you can tell them that this is just not true.

An Aside: Before posting this I wanted to verify that that quote really
was in that movie. I would have thought it would be a quote that math people,
or people who think math is hard, or math people who know math is hard,
would remember. When I googled it
all I could find was a comment on
this blog entry of Scott's
by Bill Gasarch. Not what I would call a confirmation.
It may be that my quote is not quite exact. If anyone knows for sure
(e.g., has the DVD and checks it) let me know.

I disagree; most math is very easy compared to some of the impossible things humans attempt to do all the time, like predict the behavior of the stock market, the weather, or a brain! As von Neumann once put it,

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

Keith Devlin makes the exact opposite point in his book "The Math Gene".

He makes the point that the key property of mathematics is abstraction and that humans' ability to abstract is what gave us evolutionary advantage. Therefore we (generally) all share a "math gene".

Note that he distinguishes this from the algorithm following ability associated with arithmetic and points out research showing that the two activate completely different areas of the brain. His claim is that people are turned off math by the algorithm following and as a result miss the really interesting and useful part of higher math which is the abstraction.

Along the same lines as Paul Beame's post, Vladimir Arnol'd concluded a 1997 AMS interview as follows:

Our brain has two halves: one is responsible for the multiplication of polynomials and languages, and the other half is responsible for orientation of figures in space and all the things important in real life. Mathematics is geometry when you have to use both halves.

Going back to the article, there is a continous confusion between quantities and numbers in it. What animals can recognise (even birds, at least until 3 or 4)are quantities, not numbers.

Numbers are abstract (simbolic) representations of quantities, as graphs are abstract (simbolic) representations of relations or dynamic systems are abstract representations of change.

They also say:

"In this view, numbers are a kind of technology, a man-made invention to which our all-purpose brains can adapt".

I would say that from a certain point of view the whole mathematical theories can be seen like this: a man made invention (or institution), full of free options, conventions, efficiency looking tricks...) and the fact that it is teached the platonic way make it harder than it really is.

As any other invention mathematical theories can be compared and judged according to how they fulfill their purpose. In this respect regarding theories whose purpose is to explain how the world works and why it works this way it can be said that there is still much improvement to be done and it can be done: we have now much more information about the world than ever. I would say that we have enough information...

Math is easy enough if the student works at learning and it has nothing to do with whether the teacher is a good teacher or a bad teacher. Blaming the teacher is the first thing that a student who has poor study habits does. What is hard is teaching math to students who have the mind set that the teacher's job is to entertain and make it easy.

Also there is a move afoot in the Department of Education in several universities to dumb down all math, english, and science courses. If this direction is taken then you can be sure that the USA will become a third rate country within less than thirty years. Aiming at the least motivated student will make sure that no student is motivated as we will be rewarding those who don't care about learning. Instead, get them out of the classroom before they infect the healthy students, as they are sick.

Interesting to hear TCS guys talk about Math. Do TCS people ever use integration? differentiation? Real Analysis? Functional Analysis? When you write in LaTeX, do you ever use $\int$ ? All TCS people ever think of is their polynomial time reductions! Time for TCS to leave Garey and Johnson behind and do the real math.